5 research outputs found
Sequences of odd numbers, even numbers and integer squares: gaps in frequency distributions of unit's digits of minor totals
In the paper, I consider appearance of unit's digits in minor totals of a few
integer sequences. The sequences include the sequence of even integers,
sequence of odd integers and Faulhaber polynomial at . Application of
difference tables allows predicting of which digits can appear as unit's digits
in minor totals of the sequences. Absence of some digits ("gaps" in frequency
distributions) depends often on numbering system applied. However, in case of
odd numbers' integers the gaps are found under all numbering systems with bases
from 3 to 10.Comment: 10 pages, 4 figures, 4 table
Proof of Sun's conjectures on integer-valued polynomials
Recently, Z.-W. Sun introduced two kinds of polynomials related to the
Delannoy numbers, and proved some supercongruences on sums involving those
polynomials. We deduce new summation formulas for squares of those polynomials
and use them to prove that certain rational sums involving even powers of those
polynomials are integers whenever they are evaluated at integers. This confirms
two conjectures of Z.-W. Sun. We also conjecture that many of these results
have neat -analogues.Comment: 12 pages, corrected a typo in Theorem 1.
Proof of a conjecture involving Sun polynomials
The Sun polynomials are defined by \begin{align*}
g_n(x)=\sum_{k=0}^n{n\choose k}^2{2k\choose k}x^k. \end{align*} We prove that,
for any positive integer , there hold \begin{align*}
&\frac{1}{n}\sum_{k=0}^{n-1}(4k+3)g_k(x) \in\mathbb{Z}[x],\quad\text{and}\\
&\sum_{k=0}^{n-1}(8k^2+12k+5)g_k(-1)\equiv 0\pmod{n}. \end{align*} The first
one confirms a recent conjecture of Z.-W. Sun, while the second one partially
answers another conjecture of Z.-W. Sun. We give three different proofs of the
former. One of them depends on the following congruence: Comment: 15 page
Proof of a conjecture of Z.-W. Sun on the divisibility of a triple sum
The numbers and are defined as \begin{align*}
R_n=\sum_{k=0}^{n}{n+k\choose 2k}{2k\choose k}\frac{1}{2k-1},\ \text{and}\
W_n=\sum_{k=0}^{n}{n+k\choose 2k}{2k\choose k}\frac{3}{2k-3}. \end{align*} We
prove that, for any positive integer and odd prime , there hold
\begin{align*} \sum_{k=0}^{n-1}(2k+1)R_k^2 &\equiv 0 \pmod{n}, \\
\sum_{k=0}^{p-1}(2k+1)R_k^2 &\equiv 4p(-1)^{\frac{p-1}{2}} -p^2 \pmod{p^3}, \\
9\sum_{k=0}^{n-1}(2k+1)W_k^2 &\equiv 0 \pmod{n}, \\ \sum_{k=0}^{p-1}(2k+1)W_k^2
&\equiv 12p(-1)^{\frac{p-1}{2}}-17p^2 \pmod{p^3}, \quad\text{if .}
\end{align*} The first two congruences were originally conjectured by Z.-W.
Sun. Our proof is based on the multi-variable Zeilberger algorithm and the
following observation: where .Comment: 18 page
Jacobi polynomials and congruences involving some higher-order Catalan numbers and binomial coefficients
In this paper, we study congruences on sums of products of binomial
coefficients that can be proved by using properties of the Jacobi polynomials.
We give special attention to polynomial congruences containing Catalan numbers,
second-order Catalan numbers, the sequence (\seqnum{A176898})
and the
binomial coefficients and . As an application,
we address several conjectures of Z.\ W.\ Sun on congruences of sums involving
and we prove a cubic residuacity criterion in terms of sums of the
binomial coefficients conjectured by Z.\ H.\ Sun